Let $G$ be a group acting on a finite set $X$. This also gives an action of $G$ on the subsets of $X$ of any given size, and we can ask whether this action is transitive for some specified size of subsets.
My question is first whether there is a term for this (Edit: Derek Holt has pointed out that if the induced action on subsets of size $n$ is transitive, then the action is called $n$-homogeneous).
It is clear that transitive is equivalent to $1$-homogenous, and further, one can show that if the action is $n$-homogenous for some $1\leq n\leq |X| - 1$ then it is also transitive. To see this, let $a$ and $b$ be distinct elements of $X$ and consider two subsets of $X$ of size $n$, $A = \{a,x_1,\dots,x_{n-1}\}$ and $B = \{b,x_1,\dots,x_{n-1}\}$. Let $g\in G$ be given such that $g.A = B$. Since $X$ is finite, there is some $m$ such that $g^m.a = a\not\in B$ so let $i$ be the smallest such that $g^i.a\not\in B$. If $g^j.a\in A$ then $g^{j+1}.a\in B$, so by construction we have $g^{i-1}.a\in B\setminus A = \{b\}$ so $g^{i-1}.a = b$ and the proof is done.
Another thing to note is that if $|X| = m$ then the action is $n$-homogenous if and only if it is $(m-n)$-homogenous (by looking at the corresponding action on the complements).
Now, my second question is: If the action is $n$-homogenous for some $1\leq n\leq \frac{m}{2}$ can we conclude that the action is $(n-1)$-homogenous?
For the second question: Yes, $n$-homogeneity for $n\leq|X|/2$ implies $(n-1)$-homogeneity. This is a theorem of Livingstone and Wagner. Unfortunately, I don't have quick access to the reference just now.